Question: Assume that $C$ is a positively oriented, piecewise smooth, closed curve. Let $R$ be the region enclosed by $C$. Use the circulation form of Green's theorem to rewrite $ \oint_C y^3 \, dx + e^{\sin(xy)} \, dy$ as a double integral. Choose 1 answer: Choose 1 answer: (Choice A) A $ \iint_R x\cos(xy)e^{\sin(xy)} + 3y^2 \, dA$ (Choice B) B $ \iint_R x\sin(xy)e^{\cos(xy)} - 3y^2 \, dA$ (Choice C) C $ \iint_R y\sin(xy)e^{\cos(xy)} + 3y^2 \, dA$ (Choice D) D $ \iint_R y\cos(xy)e^{\sin(xy)} - 3y^2 \, dA$ (Choice E) E Green's theorem is not necessarily applicable.
Explanation: Assume we have a two-dimensional vector field $F(x, y) = P(x, y) \hat{\imath} + Q(x, y) \hat{\jmath}$ and a piecewise smooth, simple, closed curve $C$. Let $R$ be the region enclosed by $C$. Then the circulation form of Green's theorem states that we have the equality below: $ \oint_C P \, dx + Q \, dy = \iint_R \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) dA$ Our first step should be to confirm that the given curve is compatible with using Green's theorem. Looking closely, the curve $C$ does not satisfy all the conditions of Green's theorem: the problem never specifies that $C$ is simple! Because we can't be sure whether or not the curve is simple, Green's theorem is not necessarily applicable.